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Album: Functional Analysis

90 tracks

Funct-an

Description: Functional analysis begins with a marriage of linear algebra and metric topology. These work together in a highly effective way to elucidate problems arising from differential equations. Solutions are sought in an infinite dimensional space of functions. This module paves the way by establishing the principal theorems (all due in part to the great Polish mathematician Stefan Banach) and exploring their diverse consequences. Topics to be covered will include: – norm topology and topological isomorphism; – boundedness of operators; – compactness and finite dimensionality; – extension of functionals; – weak*-compactness; – sequence spaces and duality; – basic properties of Banach algebras. Suitable for: Undergraduate students Level Four Further module materials are available for download from The University of Nottingham open courseware site: http://unow.nottingham.ac.uk/resources/resource.aspx?hid=bd32d53b-3c12-ac19-176b-d9e112731951 Dr Joel Feinstein is an Associate Professor in Pure Mathematics at the University of Nottingham. After reading mathematics at Cambridge, he carried out research for his doctorate at Leeds. He held a postdoctoral position in Leeds for one year, and then spent two years as a lecturer at Maynooth (Ireland) before taking up a permanent position at Nottingham. His main research interest is in functional analysis, especially commutative Banach algebras. Dr Feinstein has published two case studies on his use of IT in the teaching of mathematics to undergraduates. In 2009, Dr Feinstein was awarded a University of Nottingham Lord Dearing teaching award for his popular and successful innovations in this area.

Keywords: Functional analysis, Normed spaces, Banach spaces, Bounded linear operators, dual spaces, commutative Banach algebras, complete metric spaces, open mapping theorem, closed graph theorem, uniform boundedness Category: Mathematics Subcategory: Mathematics:Advanced Mathematics

Copyright: Copyright (c) University of Nottingham Language: en Explicit: no

Author: Dr Joel Feinstein Owner: The University of Nottingham Owneremail: itunesu@nottingham.ac.uk

Web page: http://www.nottingham.ac.uk/mathematics

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Tracks

Funct-an Lecture 1 - Functional Analysis, Introductory material on totally ordered sets and partially ordered sets
Funct-an Lecture 1 - ( for iPod) Functional Analysis, Introductory material on totally ordered sets and partially ordered sets
Funct-an Lecture 2 - Complete metric spaces
Funct-an Lecture 2 - (for iPod) Complete metric spaces
Funct-an Lecture 3 - revision of Metric and Topological Spaces
Funct-an Lecture 3 - (for iPod) revision of Metric and Topological Spaces
Funct-an Lecture 4 - Complete metric spaces - printed slides 7-9
Funct-an Lecture 4 - (for iPod) Complete metric spaces - printed slides 7-9
Funct-an Lecture 5a - Complete metric spaces - printed slide 9 to end of Chapter 1
Funct-an Lecture 5a - (for iPod) Complete metric spaces - printed slide 9 to end of Chapter 1
Funct-an Lecture 5b - Chapter 2: Infinite products and Tychonoff’s theorem, printed slides 13-14
Funct-an Lecture 5b - (for iPod) Chapter 2: Infinite products and Tychonoff’s theorem, printed slides 13-14
Funct-an Lecture 6a - discussion session
Funct-an Lecture 6a - (for iPod) discussion session
Funct-an Lecture 6b - discussion session part b
Funct-an Lecture 6b (for iPod) - discussion session part b
Funct-an Lecture 7 - Infinite products and Tychonoff's theorem
Funct-an Lecture 7 - (for iPod) Infinite products and Tychonoff's theorem
Funct-an Lecture 8 - The proof of Tychonoff's theorem
Funct-an Lecture 8 (for iPod) - The proof of Tychonoff's theorem
Funct-an Lecture 9a - Infinite products and Tychonoff's theorem - printed slide 26 to end of chapter
Funct-an Lecture 9a (for iPod) - Infinite products and Tychonoff's theorem - printed slide 26 to end of chapter
Funct-an Lecture 9b - Normed spaces and Banach spaces - printed slides 28-31
Funct-an Lecture 9b - (for iPod) - Normed spaces and Banach spaces - printed slides 28-31
Funct-an Lecture 10 - Normed spaces and Banach spaces - printed slides 31-34
Funct-an Lecture 10 - (for iPod) Normed spaces and Banach spaces - printed slides 31-34
Funct-an Lecture 11a - printed slides 33-34 continued, completeness of the uniform norm
Funct-an Lecture 11a - (for iPod) printed slides 33-34 continued, completeness of the uniform norm
Funct-an Lecture 11b - Interlude on pointwise convergence and uniform convergence
Funct-an Lecture 11b - (for iPod) Interlude on pointwise convergence and uniform convergence
Funct-an Lecture 12 - Normed spaces and Banach spaces - printed slides 33-34 concluded, and printed slides 35-37
Funct-an Lecture 12 - (for iPod) Normed spaces and Banach spaces - printed slides 33-34 concluded, and printed slides 35-37
Funct-an Lecture 13a - Normed spaces and Banach spaces - final discussion of Section 3.1
Funct-an Lecture 13a - (for iPod) Normed spaces and Banach spaces - final discussion of Section 3.1
Funct-an Lecture 13b - Equivalence of norms - whole section discussed, including discussion of facts needed to prove Theorem 3.8
Funct-an Lecture 13b - (for iPod) Equivalence of norms - whole section discussed, including discussion of facts needed to prove Theorem 3.8
Funct-an Lecture 14a - Equivalence of norms - recap of preliminary discussion relating to Theorem 3.8
Funct-an Lecture 14a - (for iPod) Equivalence of norms - recap of preliminary discussion relating to Theorem 3.8
Funct-an Lecture 14b - Equivalence of norms - "conclusion" of Section 3.2, including the proofs of Theorems 3.8 and 3.10
Funct-an Lecture 14b - (for iPod) Equivalence of norms - "conclusion" of Section 3.2, including the proofs of Theorems 3.8 and 3.10
Funct-an Lecture 15a - Equivalence of norms - final discussion of Section 3.2
Funct-an Lecture 15a - (for iPod) Equivalence of norms - final discussion of Section 3.2
Funct-an Lecture 15b - Linear maps - Section 3.3, printed slides 41-43
Funct-an Lecture 15b - (for iPod) Linear maps - Section 3.3, printed slides 41-43
Funct-an Lecture 16a - (for iPod) Linear maps - Conclusion of Section 3.3 including recap, & printed slide 44.
Funct-an Lecture 16a - Linear maps - Conclusion of Section 3.3 including recap, & printed slide 44
Funct-an Lecture 16b - Sequence spaces - Section 3.4,  printed slides 45-48
Funct-an Lecture 16b - (for iPod) Sequence spaces - Section 3.4,  printed slides 45-48
Funct-an Lecture 17 - Sequence spaces - Recap of Section 3.4 and proof of part of Theorem 3.15
Funct-an Lecture 17 - (for iPod) Sequence spaces - Recap of Section 3.4 and proof of part of Theorem 3.15
Funct-an Lecture 18a - Sequence spaces - Conclusion of Section 3.4
Funct-an Lecture 18a - (for iPod) Sequence spaces - Conclusion of Section 3.4
Funct-an Lecture 18b - Isomorphisms - Introduction to Section 3.5: printed slides 49-51
Funct-an Lecture 18b - (for iPod) Isomorphisms - Introduction to Section 3.5: printed slides 49-51
Funct-an Lecture 19a - Isomorphisms - Conclusion of Section 3.5: printed slides 51-54
Funct-an Lecture 19a - (for iPod) Isomorphisms - Conclusion of Section 3.5: printed slides 51-54
Funct-an Lecture 19b - Sums and quotients of vector spaces - Section 3.6, printed slides 55-57
Funct-an Lecture 19b - (for iPod) Sums and quotients of vector spaces - Section 3.6, printed slides 55-57
Funct-an Lecture 20a - Sums and quotients of vector spaces - Conclusion of Section 3.6, printed slides 57-60
Funct-an Lecture 20a - (for iPod) Sums and quotients of vector spaces - Conclusion of Section 3.6, printed slides 57-60
Funct-an Lecture 20b - Section 3.7: Dual spaces - Section 3.7, printed slides 61-64
Funct-an Lecture 20b - (for iPod) Section 3.7: Dual spaces - Section 3.7, printed slides 61-64
Funct-an Lecture 21 - Duals and Double Duals - Section 3.7, printed slides 64-65
Funct-an Lecture 21 - (for iPod) Duals and Double Duals - Section 3.7, printed slides 64-65
Funct-an Lecture 22 - Conclusion of Section 3.7 and brief introduction to Section 3.8
Funct-an Lecture 22 - (for iPod) Conclusion of Section 3.7 and brief introduction to Section 3.8
Funct-an Lecture 23 - Extensions of linear maps
Funct-an Lecture 23 - (for iPod) Extensions of linear maps
Funct-an Lecture 24 - Completions, quotients and Riesz's Lemma
Funct-an Lecture 24 - (for iPod) Completions, quotients and Riesz's Lemma
Funct-an Lecture 25 - The Weak-* Topology and the Banach-Alaoglu Theorem
Funct-an Lecture 25 - (for iPod) The Weak-* Topology and the Banach-Alaoglu Theorem
Funct-an Lecture 26 - Open Mappings and their Applications
Funct-an Lecture 26 - (for iPod) Open Mappings and their Applications
Funct-an Lecture 27 - The open mapping lemma
Funct-an Lecture 27 - (for iPod) The open mapping lemma
Funct-an Lecture 28 - part a - Recap concerning convex sets which are symmetric about 0
Funct-an Lecture 28 - part a - (for iPod) Recap concerning convex sets which are symmetric about 0
Funct-an Lecture 28 - part b - Chapter 5, printed slides 102-108, Proof of Open Mapping Theorem
Funct-an Lecture 28 - part b (for iPod) - Chapter 5, printed slides 102-108, Proof of Open Mapping Theorem
Funct-an Lecture 32 (for iPod) - Discussion session on measure theory
Funct-an Lecture 29 part a - Recap, and proof of the Closed Graph Theorem
Funct-an Lecture 29 part a - (for iPod) Recap, and proof of the Closed Graph Theorem
Funct-an Lecture 29 part b - The Uniform Boundedness Principle/Banach-Steinhaus
Funct-an Lecture 29 part b (for iPod) - The Uniform Boundedness Principle/Banach-Steinhaus
Funct-an Lecture 30 - Commutative Banach Algebras, printed slides 1-19
Funct-an Lecture 30 (for iPod)- Commutative Banach Algebras, printed slides 1-19
Funct-an Lecture 31 - Commutative Banach Algebras, printed slides 19-29
Funct-an Lecture 31 (for iPod) - Commutative Banach Algebras, printed slides 19-29
Funct-an Lecture 32 - Discussion session on measure theory